Vol. 81, No. 1, 1979

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Generalizations of the Robertson functions

Edward Jean Moulis, Jr.

Vol. 81 (1979), No. 1, 167–174

We study a class of analytic functions which unifies a number of classes previously studied, including functions with boundary rotation at most , functions convex of order ρ and the Robertson functions, i.e., functions f for which zfis α-spirallike. We obtain representation theorems for this general class, and using a simple variational formula, also obtain sharp bounds on the modulus of the second coefficient of the series expansion of these functions. Using a univalence criterion due to Ahlfors, we determine a condition on the parameters k, α, and ρ which will ensure that a function in this class is univalent. This result improves previously published results for various subclasses and is sharp for the class of functions f for which zfis α-spirallike of order ρ.

Mathematical Subject Classification 2000
Primary: 30C45
Secondary: 30C50
Received: 14 March 1978
Published: 1 March 1979
Edward Jean Moulis, Jr.